Historically the subject is traced back to Kerekjarto’s classification of non-compact surfaces in 1923, but its emergence as an important tool in geometric topology came with Larry Siebenmann’s work in 1965.

Suppose MM is a smoothmanifold with boundary, ∂M\partial M, then M∖∂MM\setminus \partial M is an open manifold. Now suppose someone gives us an open manifold NN, is it possible to detect if there is a compact manifold MM, with M∖∂≅NM\setminus \partial \cong N. Siebenmann showed that certain conditions on the ends of NN were necessary and that there were obstructions if the dimension of MM was greater than 5.

Its potential importance of proper homotopy, for example for physical applications, comes from the fact that the phenomena it studies include the limiting behaviour of the system.

Intuitions

The basic hypothesis will be that XX will be a connected and locally connected compact Hausdorff space. It will usually be σ\sigma-compact, i.e., there will be an increasing sequence, {Kn}\{K_n\}, of compact subspaces with each KnK_n in the interior of Kn+1K_{n+1} and such that

We will be interested in the homotopy of such spaces ‘out towards its ends’

Ends

To illustrate the idea of the ends of a space XX, we note that naively ℝ\mathbb{R} has two ends, ∞\infty and −∞-\infty, whilst ℝ2\mathbb{R}^2 has only one as it is S2∖{∞}S^2\setminus \{\infty\}, (but that is vague!).

This is an inverse system or pro-object in the category of spaces. Applying the connected component functor, π0\pi_0, to this system of spaces gives π0(X)\pi_0(X), and, classically, one takes the limit of this to get

e(X)=limπ0ε(X),e(X) = lim \pi_0\varepsilon(X),

the set of ends of XX. In general, e(X)e(X) would be given the inverse limit topology?, to preserve more of the information coming from its construction. This space is the space of (Freudenthal) ends of XX. It is a profinite space.

Examples

Let X8X_8 be the figure eight space, the one-point union of two circles, and let XX be its universal cover. This is an infinite ‘thorn bush’. It has infinitely many ends and

Proper maps

The assignment sending XX to e(X)e(X) cannot be functorial on the category of spaces and continuous maps, since the contracting map ℝ→{0}\mathbb{R}\to \{0\} is continuous, e(ℝ)e(\mathbb{R}) is {−∞,∞}\{-\infty, \infty\}, whilst e({0})e(\{0\}) is empty since {0}\{0\} is compact. The problem is that continuity is really about inverse images (inverse image of open is open), but the inverse image mapping does not preserve compactness (as in the example!).

Definition (recall)

A map f:X→Yf:X\to Y is a proper map if for each subset KK compact in YY then f−1(K)f^{-1}(K) is compact in XX.

End spaces continued

If f:X→Yf:X\to Y is proper, then it induces a pro-morphism

ε(f):ε(X)→ε(Y)\varepsilon(f) : \varepsilon(X)\to \varepsilon(Y)

and hence a continuous map of the end spaces

e(f):e(X)→e(Y),e(f):e(X)\to e(Y),

and ee becomes a functor from some category ProperProper of spaces and proper maps to StoneStone, the category of profinite spaces / Stone spaces.

Proper homotopy

Proposition

For any space XX, the natural inclusions of XX into X×IX \times I, ei(x)=(x,i)e_i(x) = (x,i), i=0,1i = 0,1 together with the projection maps from X×IX\times I to XX, are proper maps.

Corollary

We call the corresponding notion of homotopy, ‘proper homotopy’. We get a ‘Proper category’ and an associated ‘proper homotopy category’, which we will denote Ho(Proper)Ho(Proper). If we are restricting to σ\sigma-compact spaces we may write ProperσProper_\sigma, and so on.

Germs at ∞\infty

Although a proper map f:X→Yf: X\to Y will induce a continuous ε(f)\varepsilon(f), on the end spaces, it is clear that ff does not need to be defined on the whole of XX for this to work, as ε\varepsilon encodes behaviour ‘out towards ∞\infty’. This leads to the notion of a ‘germ at ∞\infty’.

Suppose XX is locally compact Hausdorff and A⊂XA\subset X. The inclusion j:A→Xj: A\to X is ‘cofinal’ if the closure of X∖AX\setminus A is compact. Note that a cofinal inclusion is proper and induces an isomorphism ε(A)≅ε(X)\varepsilon(A)\cong \varepsilon(X). Let Σ\Sigma be the class of all cofinal inclusions in ProperProper and let Proper∞=Proper[Σ−1]Proper_\infty = Proper[\Sigma^{-1}], the category obtained by formally inverting the cofinal inclusions.

Definition

This category is called the proper category at ∞\infty.

Note that (Proper,Σ)(Proper,\Sigma) admits a calculus of right fractions, so any morphism from XX to YY in Proper∞Proper_\infty can be represented by a diagram

X←jA→fY,X\stackrel{j}{\leftarrow} A \stackrel{f}{\rightarrow}Y,

with jj a cofinal inclusion, i.e., ff is defined on some ‘neighbourhood of the end of XX’.

represent the same germ if f′|A=f′′|Af^\prime | A = f^{\prime\prime}|A for some cofinal subspace AA with A′∪A′′⊂AA^\prime \cup A^{\prime\prime}\subset A.

There is also a homotopy category Ho(Proper∞)Ho(Proper_\infty)

The end space e(X)e(X) is a Stone space so is Max(R)Max(R) for some Boolean algebra RR (Stone duality) In the 1960s someone (Goldman?) looked at a ring, RR, of ‘almost continuous functions’ from XX to ℤ/2ℤ\mathbb{Z}/2\mathbb{Z}, that gave the right e(X)e(X). Can this idea help integrate better the ideas of proper homotopy etc. with modern methods of algebraic geometry?

Proper analogues of the fundamental group

The end space behaves a bit like a π0\pi_0 and usually spaces will have many ends, so are not ‘connected at infinity’. If we try to do a fundamental group or groupoid analogue, this means life will get more complicated. We will try with the assumption of a space having a single end for simplicity! We will also assume XX is σ\sigma-compact.

First attempt

We could try defining π1(ε(X))\pi_1(\varepsilon(X)) as a progroup, then taking its limit. For this we would take {Kn}\{K_n\} an exhausting increasing sequence of compact subsets and setting Ui=X∖KiU_i = X\setminus K_i, pick a base point xix_i in each UiU_i, and we will get groups π1(Ui,xi)\pi_1(U_i,x_i). We however need induced homomorphisms π1(Ui+1,xi+1)→π1(Ui,xi)\pi_1(U_{i+1},x_{i+1})\to \pi_1(U_i,x_i), and for this we have to choose an arc in UiU_i from xi+1x_{i+1} to xix_i. We can combine these to get a base ray, rather than a base point, that is, we need a proper map, α:[0,∞)→X\alpha : [0,\infty)\to X. With that we do get an inverse sequence of groups, but there are problems. What is the dependence of the inverse system on the choice of α\alpha?

Let XX be an infinite cylinder with an infinite string of circles attached via a proper ray α:[0,∞)→X\alpha: [0,\infty) \to X. The space has just one ‘end’ but you can choose different ways of going from π1(Ui+1,xi+1)\pi_1(U_{i+1},x_{i+1}) to π1(Ui,xi)\pi_1(U_i,x_i) for fairly obvious choices of base points such that the limit groups of the resulting two inverse systems are non-isomorphic! (In the survey listed below, this example is examined in detail, and one of the limits is a free group on one element, the other is trivial! Definitely non-isomorphic!)

This means that limπ1(ε(X))lim \pi_1(\varepsilon(X)) is not an invariant of the end. This phenomenon is linked to the fact that π1(ε(X))\pi_1(\varepsilon(X)) does not satisfy the Mittag-Leffler condition for either choice of the base rays.

Waldhausen boundary

If XX and YY are locally compact Hausdorff spaces, there is no obvious candidate for a space of proper maps from XX to YY, but one can form a simplicial set ℙ(X,Y)\mathbb{P}(X,Y) with ℙ(X,Y)n=Proper(X×Δn,Y)\mathbb{P}(X,Y)_n = Proper(X\times \Delta^n,Y), which acts as if it was the singular complex of the mythical space of proper maps from XX to YY.

Definition

The Waldhausen boundary of XX is the simplicial set ℙ([0,∞),X)\mathbb{P}([0,\infty),X).

There is an epimorphism from π0(ℙ([0,∞),X))\pi_0(\mathbb{P}([0,\infty),X)) to e(X)e(X).

In the example above of the cylinder with the string of circles attached, π0(ℙ([0,∞),X))\pi_0(\mathbb{P}([0,\infty),X)), is uncountable, and π1(ℙ([0,∞),X))\pi_1(\mathbb{P}([0,\infty),X)) maps onto limSlim S.

When XX has a single end and π0(ε(X))\pi_0(\varepsilon(X)) is Mittag-Leffler, then π0(ℙ([0,∞),X))\pi_0(\mathbb{P}([0,\infty),X)), is a single point, i.e. all possible base rays are properly homotopic.

Second attempt

Even if we did not have the above difficulty with the limit groups, we would still have the problem that, as the limit functor is not exact, the resulting limiting homotopy groups would not be that well behaved. There would not be any general long exact sequence results (just as with Čech homology). There is at least one possible replacement for those limiting homotopy groups, but first we note that it is not appropriate to base any such things at a point, rather we should be using a base ray as was discussed above.

One fairly obvious attempt to define a ‘fundamental group’ for XX, based at a proper ray α:[0,∞)→X\alpha: [0,\infty) \to X, would be to note that α\alpha gives ℙ([0,∞),X)\mathbb{P}([0,\infty),X) a base point so we could look at π1(ℙ([0,∞),X),α)\pi_1(\mathbb{P}([0,\infty),X),\alpha) and more generally at πn(ℙ([0,∞),X),α)\pi_n(\mathbb{P}([0,\infty),X),\alpha), and we will denote these groups by π̲̲n(X,α)\underline{\underline{\pi}}_n(X,\alpha).

There are variants ‘at infinity’ of both the Waldhausen boundary and these groups, otained using germs instead of proper maps. These will be denoted with a ∞\infty as a super- or suffix on the above notation.

Brown–Grossman homotopy groups

(Once over lightly here, more details at Brown-Grossman homotopy group.) In fact there is another different way of looking at these groups, which has a more geometric feel to it. Historically these groups were not the first successful attempt. This was due to Ed Brown and uses strings of spheres. These are examples of spherical objects and the resulting ‘groups’ (better thought of as ‘Π𝒜\Pi_\mathcal{A}-algebras’) have a rich structure. They are discussed at Brown-Grossman homotopy groups. They do link with the above homotopy groups of the Waldhausen boundary, which are called the Steenrod homotopy group?s, (for reasons that will be explained there).